Modelling Communication Dynamics
in Social Networks
Amber Tomas
September 28, 2011
Key Points
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Aim is to understand the factors influencing communication
patterns
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Model communications as events on edges of a network
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Allow for factors influencing communication to change over
time
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Jointly model events on edges and nodes
Questions:
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What are the patterns of communication?
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How can we explain them?
Questions:
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What are the patterns of communication?
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How can we explain them?
Statistical modelling allows for:
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Analysis of behaviour (testing hypotheses)
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Improved understanding of communication systems
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Monitoring of communication patterns for changes
Network Event Data
At any time t
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There exists a set of actors who may communicate
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There exists a set of possible communications between pairs
of actors
An “event” is an instantaneous communication between two actors.
Network Event Data
Network Event Data
Network Event Data
Network Event Data
Network Event Data
Network Event Data
Network Event Data
Every event is recorded as a tuple
(a, b, k, t):
a − sender
b − receiver
k − type of event
t − time of event
Every event is recorded as a tuple
(a, b, k, t):
a − sender
b − receiver
k − type of event
t − time of event
A dataset consists of
1. A sequence of recorded
events
2. Information about each of
the actors
Example - Enron Email Corpus
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1.8 million emails sent from employees of Enron between
November 1998 and September 2002
Data includes
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Time stamp
sender
recipients
type of recipient (to, cc, bcc)
subject
message content
Models for Event Data
Butts (2008) proposed a “relational event framework” for
modelling network event data.
This models the rate of events between actors conditional on event
history and actor characteristics.
Events are assumed to be independent conditional on the past
event history.
Models for Event Data
The likelihood of the event history up to the most recent event is
given by
##
" t "
Y
Y
S(ti − ti−1 |i , e1:t−1 )
p(e1:t ) =
h(ti |ei , e1:i ) ×
i=1
i
Models for Event Data
The likelihood of the event history up to the most recent event is
given by
##
" t "
Y
Y
S(ti − ti−1 |i , e1:t−1 )
p(e1:t ) =
h(ti |ei , e1:i ) ×
i=1
i
It is assumed the hazard is constant given past history, so
h(t) = λ
S(∆t) = e −λ∆t
Models for Event Data
The rate function is modelled as
n
o
λ(et , e1:t−1 , X , θ) = exp λ0 + θT u(et , e1:t−1 , X )
Models for Event Data
The rate function is modelled as
n
o
λ(et , e1:t−1 , X , θ) = exp λ0 + θT u(et , e1:t−1 , X )
Considering only the ordering of data leads to the likelihood
p(e1:t |θ, X ) =
t
Y
i=1
exp θT u(et , e1:t−1 , X )
P
T
i exp {θ u(i , e1:t−1 , X )}
Models for Event Data
The rate function is modelled as
n
o
λ(et , e1:t−1 , X , θ) = exp λ0 + θT u(et , e1:t−1 , X )
Considering only the ordering of data leads to the likelihood
p(e1:t |θ, X ) =
t
Y
i=1
exp θT u(et , e1:t−1 , X )
P
T
i exp {θ u(i , e1:t−1 , X )}
An important aspect of fitting this model is choosing the statistics
u(et , e1:t−1 , X ).
Choice of Statistics
1. Characteristics of sender a and receiver b
2. Tendency for a to send messages
3. Tendency for b to receive messages
4. Tendency for communications between a and b
5. Communications involving joint contacts of a and b
Choice of Statistics
1. Characteristics of sender a and receiver b
2. Tendency for a to send messages (node)
3. Tendency for b to receive messages (node)
4. Tendency for communications between a and b (dyad)
5. Communications involving joint contacts of a and b (triad etc)
Including network statistics allows for incorporation of complex
dependencies.
Some Possible Statistics
To simplify computation, an event network is used to compute
statistics.
The weight of an edge is an exponentially weighted aggregation of
past events (Brandes et al).
Some Possible Statistics
Model Fitting
Model fitting can proceed as follows:
1. For every event
1.1 calculate relevant statistics
1.2 calculate contribution to likelihood
1.3 Update weighted event network
2. Estimate parameters using maximum likelihood
Model Fitting
Model fitting can proceed as follows:
1. For every event
1.1 calculate relevant statistics
1.2 calculate contribution to likelihood
1.3 Update weighted event network
2. Estimate parameters using maximum likelihood
Alternatively, it is natural to update the parameter estimates
sequentially.
Model Fitting
The main difficulty is in specifying appropriate statistics, and
correctly interpreting the fitted model.
Example - Enron Data
ENRON
Example - Enron Data
ENRON
SUCCESSFUL
COMPANY
Example - Enron Data
ENRON
SUCCESSFUL
COMPANY
REALITY
Example - Enron Data
ENRON
SUCCESSFUL
COMPANY
REALITY
Accounting
Shenanigans
Example - Enron Data
Considered a subset of email messages
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Only consider addresses ending in @enron.com
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Only consider messages sent to <= 5 receivers
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Only consider actors who send and receive at least 2 messages
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Leaves 134 actors
Aggregated Data
Aggregated Data
Aggregated Data
Aggregated Data
800
400
600
count
300
200
0
200
100
0
count
400
1000
Rate of Sending
1
4
7
10
14
time
18
22
M
T
W
T
day
F
S
S
60
40
20
0
receiver
80
100
120
Candidate Events
0
200
400
600
800
time (days)
1000
1200
Choice of Statistics
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300
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Mark Taylor
200
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50
100
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0
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John Lavorato
●
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0
number sent
400
Tana Jones
Jeff Dasovich
Richard
Shapiro
●
●
●
●
●
●
100
●
Louise Kitchen
●
150
200
250
number received
300
350
Choice of Statistics
●●
300
●
Mark Taylor
200
●
50
0
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0
●
John Lavorato
●
●
100
number sent
400
Tana Jones
Jeff Dasovich
Richard
Shapiro
●
●
●
●
●
●
100
●
Louise Kitchen
●
150
200
250
300
350
number received
Need to control for actor tendency to send/receive messages
Choice of Statistics
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“Inertia” wab
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“Reciprocity” wba
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Sender “out-degree”
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Triad effects:
P
h
a
wah
h
b
Transitivity
h
a
h
b
Shared Triad
a
b
Choice of Statistics
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role (manager, lawyer, executive, other)
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role homophily
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new member
50
Results
0
10
20
E[βi]
30
40
inertia
reciprocity
sender.out.degree
receiver.out.degree
sender.in.degree
receiver.in.degree
0
1000
2000
3000
time
4000
5000
6000
Results
0
10
20
E[βi]
30
40
50
inertia
reciprocity
sender.out.degree
receiver.out.degree
sender.in.degree
receiver.in.degree
0
1000
2000
3000
time
4000
5000
6000
Results
−10
0
10
E[βi]
20
30
inertia
reciprocity
sender.out.degree
receiver.out.degree
sender.in.degree
receiver.in.degree
transitivity
shared.triad
joint.triad
0
1000
2000
3000
time
4000
5000
6000
Example - Hospital Acquired Infections
Data is available for a large hospital network, including
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Patient ward transfers (time, from ward, to ward, patient
condition)
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Tests for hospital acquired infections (bug, time of test, time
of result, result, patient info)
Example - Hospital Acquired Infections
Data is available for a large hospital network, including
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Patient ward transfers (time, from ward, to ward, patient
condition)
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Tests for hospital acquired infections (bug, time of test, time
of result, result, patient info)
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14 000 transfers (11 000 discharged or died)
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22 000 bug tests
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2 700 bug positives
Example - Hospital Acquired Infections
Data is available for a large hospital network, including
I
Patient ward transfers (time, from ward, to ward, patient
condition)
I
Tests for hospital acquired infections (bug, time of test, time
of result, result, patient info)
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14 000 transfers (11 000 discharged or died)
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22 000 bug tests
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2 700 bug positives
We might be interested in the joint dynamics of infections and
ward transfers.
Hospital Data
M
K
D
F
A
J
H
O
N
C
B
E
G
P
I
L
Hospital Data
Can model the joint dynamics as
p(e1:E , v1:V |θ, γ) = p(e1:E |θ)p(v1:V |γ),
where
p(v1:V |γ) =
Y
i
exp γ T u 0 (et<i , vt<i )
P
T 0
ν exp {γ u (et<i , νt<i )}
Hospital Data
Statistics might include
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number recently admitted to ward
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number on ward
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infection history of ward
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infection history of source wards
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transfer patterns and rates between wards
Conclusions
Network event data is becoming more common. Analysis can help
improve understanding of the factors driving the communication
dynamic.
Conclusions
Network event data is becoming more common. Analysis can help
improve understanding of the factors driving the communication
dynamic.
Possible applications include
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Business and emergency communications
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political dynamics
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personal communications
New studies should be designed with the possibilities for network
analysis in mind.
Network Models
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The appropriate model for any data set will depend on what
we are interested in.
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Suppose interested in the probability of an edge between two
actors, given personal characteristics, network data.
The probability of a tie between actors A and B might depend
on
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which other ties are present (degree, reciprocity, transitivity,
etc)
actor characteristics (age, sex, role, etc)
Network Models - ERGM
P(X = x) ∝ exp{
X
βk sk (x)},
k
where β is a vector of parameters and s(x) is a vector of statistics.
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implies edge probabilities
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allows testing of hypotheses regarding significance of actor or
network statistics
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normalizing constant causes computational difficulties
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requires a single observation of the network
ERGMs and Event Data
We could aggregate all events to produce a “communication
network”, i.e. who had communications with whom at any point in
time.
This allows investigation of questions such as
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Is transitivity a significant influence on the structure of the
network?
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How many contacts do people have?
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Are communications reciprocated?
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Are communications more likely among actors of similar
gender/age.ethnicity etc?
But this tells us nothing about the temporal aspects.
Network Models - SAOMs
Stochastic actor-oriented network models (SAOMs) have been
developed to model longitudinal network data.
We could aggregate event data in sequential time intervals to
produce such data.
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Allows testing of temporal hypotheses, eg did A become a
manager because he was communicating with manager B, or
did A only begin to communicate with B after obtaining a
promotion independently?
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Results may be sensitive to the periods of aggregation
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Tells us little about commnuication rates or patterns over
short time intervals.